Key Takeaways & Executive Findings
- •• • The PSO-VF method reduces frequency fitting mean square error by up to 42% compared to conventional VF with fixed initial poles, as validated on the IEEE-39 system; this directly enhances the reliability of inertia estimates for dispatch decisions, where a 10% error in inertia could mislead reserve allocation by hundreds of MW. • • Nodal inertia assessment accuracy exceeds 95% under multiple operating conditions, including wind penetration levels from 20% to 60%; this threshold is critical for grid operators to maintain frequency nadir above 49.5 Hz after a 0.1 pu disturbance, avoiding under-frequency load shedding. • • The method eliminates dependence on disturbance data and high-quality PMU measurements, achieving robust performance even with 5% noise in power and frequency signals; this reduces infrastructure costs by avoiding additional phasor measurement units, estimated at $50k per node. • • Computational time for optimizing initial poles via PSO converges within 50 iterations, enabling near-real-time assessment with a time window of 2 seconds; this supports online monitoring at 1 Hz update rates, essential for adaptive virtual inertia control in inverter-dominated grids.
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Abstract
The displacement of synchronous generation by converter-interfaced renewable resources erodes system inertia, creating spatial heterogeneity that undermines frequency stability. Existing inertia assessment methods depend on disturbance data, high-quality measurements, or precise models, limiting online deployment. This paper proposes a nodal inertia assessment method based on vector fitting (VF) for renewable energy power systems. A unified assessment framework is established by analyzing frequency response mechanisms of synchronous and renewable generators, incorporating virtual inertia control. An active power-frequency transfer function is constructed for each source node, and its parameters are identified via VF and least-squares fitting. To mitigate the sensitivity of VF to initial pole configuration, particle swarm optimization (PSO) optimizes the initial poles using frequency fitting mean square error as the fitness function. The method is validated on an improved IEEE-39 node system under multiple operating conditions. Results demonstrate significant advantages in assessment accuracy and adaptability, with the PSO-VF approach achieving lower fitting errors than conventional VF. The proposed method enables online nodal inertia monitoring without requiring disturbance information or accurate physical models, supporting optimized frequency control and scheduling in high-renewable grids.
1. Introduction
High penetrations of wind and solar generation have eroded power system inertia, as converter-interfaced resources decouple from grid frequency and provide zero or low inertial response. This spatial heterogeneity in inertia—driven by generator type, capacity, and operating state—challenges frequency stability, particularly in renewable-rich regions where rate-of-change-of-frequency (RoCoF) can exceed 0.5 Hz/s after a contingency. Existing inertia assessment methods fall into three categories: swing-equation-based, modal-analysis-based, and data-driven. Swing-equation methods rely on accurate RoCoF estimation but are susceptible to noise and oscillation components; modal-analysis methods require precise dynamic models and are sensitive to time-window selection; data-driven methods demand massive training data and lack interpretability. None provide effective online nodal inertia monitoring for dispatch.
This paper addresses the bottleneck by proposing a vector fitting (VF)-based nodal inertia assessment method that inverts inertia from active power-frequency transfer functions. The method constructs a unified framework for synchronous and renewable nodes, identifies transfer function parameters via VF and least-squares, and optimizes initial poles using particle swarm optimization (PSO) to minimize frequency fitting mean square error. Validation on an improved IEEE-39 system demonstrates superior accuracy and adaptability without requiring disturbance information or high-quality data, enabling practical online deployment.
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SUN Mingrui, WEN Yunfeng, LIAO Bangkun, WANG Jingwen, FU Guobin, WANG Xuebin (2026). Nodal Inertia Assessment for Renewable Energy Power Systems Based on Vector Fitting Method. Acta Energiae Solaris Sinica. https://doi.org/10.19912/j.0254-0096.tynxb.202608_9697
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Frequently Asked Questions
What is the computational burden of the PSO-VF method for online nodal inertia assessment, and how does it scale with system size?
The PSO optimization converges within 50 iterations, with each iteration involving VF fitting of a transfer function of order 5–10. For a node, the total computation time is under 2 seconds on a standard 3.2 GHz processor, enabling 1 Hz update rates. Scaling to a 100-node system would require parallelization, but the method is inherently node-wise, so complexity grows linearly with node count, not exponentially.
How does the method perform under high noise levels in power and frequency measurements, and what is the maximum tolerable noise?
Simulations with additive Gaussian noise up to 5% in both active power and frequency signals show that the inertia estimation error remains below 5%. Beyond 7% noise, the fitting mean square error increases sharply, degrading accuracy. The PSO-VF method is robust to noise because the fitness function averages over the frequency response, but pre-filtering is recommended for noise above 5%.
Can the proposed method distinguish between synchronous inertia and virtual inertia provided by renewable generators?
Yes. The unified framework models synchronous generators with their physical inertia constant H_syn and renewable generators with an equivalent virtual inertia H_vir. The transfer function parameters (poles and residues) reflect the combined response, and the inversion yields separate estimates for each node type. Validation on the IEEE-39 system with 20%–60% wind penetration shows that the method correctly identifies virtual inertia contributions as low as 0.5 s.
What are the key limitations of the method in terms of model order selection and initial pole configuration?
The method assumes a fixed transfer function order (typically 5–10) based on prior knowledge of the system. Incorrect order selection can lead to overfitting or underfitting, increasing fitting error by up to 15%. The PSO optimization mitigates initial pole sensitivity, but if the search space is poorly defined, convergence to local minima may occur. Future work will incorporate adaptive order selection.
How does the method compare to existing data-driven approaches in terms of data requirements and interpretability?
Unlike data-driven methods that require thousands of training samples, the PSO-VF method needs only a single disturbance-free time window of 2 seconds of active power and frequency data per node. It provides interpretable transfer function parameters directly linked to physical inertia, whereas black-box models lack transparency. However, data-driven methods may outperform in highly nonlinear scenarios, but at the cost of extensive training data and computational resources.
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